## Upper bound for the number of zeros of a meromorphic function outside a vertical line and applications. (Majoration du nombre de zéros d’une fonction méromorphe en dehors d’une droite verticale et applications.)(French, English)Zbl 1203.11059

Let $$a\in{\mathbb R}$$, $$h(s)$$ a meromorphic function in $${\mathbb C}$$, real on $${\mathbb R}$$, with only finitely many poles, finitely many zeros in the half-space $$\sigma>a$$, holomorphic and non-zero on the critical line $$\sigma=a$$. Let $$f(s)=f^{\pm}(s)=h(s)\pm h(2a-s)$$. Suppose that the function $$F(s)=\frac{h(2a-s)}{h(s)}$$ satisfies
(i) for every $$\eta>0$$ there exists $$\sigma_0=\sigma_0(\eta)>a$$ such that $$|F(s)|<\eta$$ if $$\sigma\geq \sigma_0$$ ($$s=\sigma+i\tau$$);
(ii) for every $$\varepsilon>0$$ and $$\sigma_0>a$$ there exists a sequence $$(T_n)_n$$ such that $$\lim_{n\to\infty}T_n=\infty$$ and $$|F(s)|<e^{\varepsilon |s|}$$ for $$a\leq \sigma\leq \sigma_0,|\tau|=T_n,n\geq 1$$.
The author proves that all zeros of $$f(s)$$ up to a finite number lie on the line $$\sigma=a$$ and are simple. He studies translations of the Riemann zeta-function and $$L$$-functions, and integrals of Eisenstein series, among others.

### MSC:

 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses

PARI/GP
Full Text:

### References:

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